Reflexive sets

By the axiom of foundation, there are no reflexive sets. In ill-founded set? theory, however, there may be many reflexive sets.

A Quine atom is a minimally reflexive set:

X={X}. X = \{X\} .

In Peter Aczel's ill-founded set theory, there is a unique Quine atom. On the other hand, by exempting Quine atoms (and only Quine atoms) from the axiom of foundation, one obtains a theory of pure sets equivalent to well-founded material sets with urelements.

Last revised on January 7, 2018 at 20:54:19.
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